At the first magic angle \(\theta =1.05^{\circ }\)  and the parameters listed in Table. REF , \(L \approx 0.32l\)  for Ref. [1]} and \(L \approx 0.65l\)  for Ref. [2]}. Therefore, the lattice relaxation is expected to be stronger for Ref.[1]}, with a larger increase of AB/BA stacking regions and a larger decrease of AA stacking regions. As such, the contribution of higher Fourier harmonics to the relaxation is larger, as confirmed by the value of \(\tilde{\epsilon }_3\)  in Table. REF  defined via Eqs.REF -REF  and obtained from minimizing \(U_E+U_B\)  defined in Eqs.(REF ) and (REF ). For Ref.[2]}, the lattice relaxation is weaker and smoother; it is dominated by the lowest harmonic terms, i.e. \(\tilde{\epsilon }_1\) . As we show in the later sections, an important consequence of this lattice relaxation for the electronic structure is that one must go beyond the Bistritzer-MacDonald model [5]} and include the inter-layer tunneling terms with a larger momentum transfer than just the first shell in order to obtain an accurate description of the magic angle narrow bands.
